Q.
Two equal mass $m$ and $m$ are hung from balance whose scale pans differ in vertical height by $h$. Calculate the error in weighing. If any, in terms of density of earth $\rho$.
A) $\frac{2}{3}\pi\rho R^{3}Gm$
B) $\frac{8}{3}\pi\rho Gmh$
C) $\frac{8}{3}\pi\rho R^{3}Gm$
D) $\frac{4}{3}\pi\rho Gm^{2}h$
Explanation
(b) As with height $g$ varies as
$g^{''} = \frac{g}{\lbrack 1 + h/R\rbrack^{2}\ } = g\left\lbrack 1 - \frac{2h}{R} \right\rbrack$
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and in according with figure $h_{1} > h_{2},$ so $W_{1}$ will be lesser than $W_{2}$ and
$W_{2} - W_{1} = mg_{2} - mg_{1} = 2mg\left\lbrack \frac{h_{1}}{R} - \frac{h_{2}}{R} \right\rbrack$
or $W_{2} - W_{1} = 2m\frac{GM}{R^{2}}\frac{h}{R}$
$\left\lbrack as\ g = \frac{GM}{R^{2}}\ and\ \left( h_{1} - h_{2} \right) = h \right\rbrack$
or $W_{2} - W_{1} = \frac{2mhG}{R^{3}}\left( \frac{4}{3}\pi R^{3}\rho \right)$
$= \frac{8}{3}\ \pi\rho Gmh\ \left\lbrack as\ M = \frac{4}{3}\pi R^{3}\rho \right\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.